Solve the equation. $\dfrac{dy}{dx}=4x^3y-6x^2y$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=e^{x^4-2x^3}+C$ (Choice B) B $y=\pm\sqrt{x^4-2x^3+C}$ (Choice C) C $y=Ce^{x^4-2x^3}$ (Choice D) D $y=C\sqrt{x^4-2x^3}$
Explanation: We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=4x^3y-6x^2y \\\\ \dfrac{dy}{dx}&=y(4x^3-6x^2) \\\\ \dfrac{1}{y}\,dy&=(4x^3-6x^2)\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} \dfrac{1}{y}\,dy&=(4x^3-6x^2)\,dx \\\\ \int \dfrac{1}{y}\,dy&=\int (4x^3-6x^2)\,dx \\\\ \ln|y|&=x^4-2x^3+C_1 \\\\ e^{\ln|y|}&=e^{x^4-2x^3+C_1} \\\\ |y|&=e^{x^4-2x^3}e^{C_1} \\\\ y&=Ce^{x^4-2x^3} \end{aligned}$ [Where did we get C?] Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=Ce^{x^4-2x^3} $